A dynamic variant of the battle of the sexes

Abstract

This paper proposes a differential game to introduce dynamic interactions into the well known battle of the sexes: the husband prefers one activity (‘boxing’) and the wife another (‘ballet’). Although the game is played presumably non-cooperatively, both are interested and willing to invest into the development and maintenance of their personal relationship which is only possible if they spend some time together. We show for the symmetric version of the game that non-cooperative commitment strategies (i.e. open-loop strategies) lead to more ‘harmony’ than linear-Markov-perfect strategies where egoistic behaviour is much more pronounced. These linear-Markov-perfect strategies constrain the set of smooth, stable but nonlinear-Markov strategies in such a manner that all these nonlinear-Markov-perfect strategies lead to less egoism and more harmony. Furthermore, particular nonlinear strategies may induce two steady states (depending on history) where one of the two steady states may result in a level of harmony that exceeds the cooperative outcome; under these circumstances, non-cooperation does not harm the public good harmony, in stark contrast to a play in linear-Markov-perfect strategies. Finally, numerical simulations of asymmetric situations confirm the results that were analytically obtained for the symmetric game.